1. Field of the Invention
The invention pertains to the field of mathematical analysis and modeling, nonlinear programming and optimization technology. More particularly, the invention pertains to dynamical methods for systematically obtaining local optimal solutions, as well as, the global optimal solution of continuous as well as discrete optimization problems.
2. Description of Related Art
A large variety of the quantitative issues such as decision, design, operation, planning, and scheduling abound in practical systems in the sciences, engineering, and economics that can be perceived and modeled as either continuous or discrete optimization problems. Typically, the overall performance (or measure) of a system can be described by a multivariate function, called the objective function. According to this generic description, one seeks the best solution of an optimization problem, often expressed by a real vector, in the solution space which satisfies all stated feasibility constraints and minimizes (or maximizes) the value of an objective function. The vector, if it exists, is termed the globally optimal solution. For most practical applications, the underlying objective functions are often nonlinear and depend on a large number of variables; making the task of searching the solution space to find the globally optimal solution very challenging. The primary challenge is that, in addition to the high-dimension solution space, there are many local optimal solutions in the solution space; the globally optimal solution is just one of them and yet both the globally optimal solution and local optimal solutions share the same local properties.
In general, the solution space of an optimization problem has a finite (usually very large) or infinite number of feasible solutions. Among them, there is one and, only one, global optimal solution, while there are multiple local optimal solutions (a local optimal solution is optimal in a local region of the solution space, but not in the global solution space.) Typically, the number of local optimal solutions is unknown and it can be quite large. Furthermore, the values of an objective function at local optimal solutions and at the global optimal solution may differ significantly. Hence, there are strong motivations to develop effective methods for finding the global optimal solution.
We next discuss the discrete optimization problem. The task of solving discrete optimization problems is very challenging. They are generally NP-hard (No solution algorithm of polynomial complexity is known to solve them). In addition, many discrete optimization problems belong to the class of NP-complete problems for which no efficient algorithm is known. A precise definition of NP-complete problems is available in the literature. Roughly speaking, NP-complete problems are computationally difficult; any numerical algorithm would require in the worst case an exponential amount of time to correctly find the global optimal solution.
One popular approach to attack discrete optimization problems is to use the class of iterative improvement local search algorithms [1]. They can be characterized as follows: start from an initial feasible solution and search for a better solution in its neighborhood. If an improved solution exists, repeat the search process starting from the new solution as the initial solution; otherwise, the local search process will terminate. Local search algorithms usually get trapped at local optimal solutions and are unable to escape from them. In fact, the great majority of existing optimization techniques for solving discrete optimization problems usually come up with local optimal solutions but not the global optimal one.
The drawback of iterative improvement local search algorithms has motivated the development of a number of more sophisticated local search algorithms designed to find better solutions by introducing some mechanisms that allow the search process to escape from local optimal solutions. The underlying ‘escape’ mechanisms use certain search strategies to accept a cost-deteriorating neighborhood to make escape from a local optimal solution possible. These sophisticated local search algorithms include simulated annealing, genetic algorithm, Tabu search and neural networks.
However, it has been found in many studies that these sophisticated local search algorithms, among other problems, require intensive computational efforts and usually can not find the globally optimal solution.
In addition, several effective methods are developed in this invention for addressing the following two important and challenging issues in the course of searching for the globally optimal solution:                (i) how to effectively move (escape) from a local optimal solution and move on toward another local optimal solution; and        (ii) how to avoid revisiting local optimal solutions which are already known.        
In the past, significant efforts have been directed towards attempting to address these two issues, but without much success. Issue (i) is difficult to solve and the existing methods all encounter this difficulty. Issue (ii), related to computational efficiency during the course of search, is also difficult to solve and, again, the majority of the existing methods encounter this difficulty. Issue (ii) is a common problem which degrades the performance of many existing methods searching for the globally optimal solution: re-visitation of the same local optimal solution several times; this action indeed wastes computing resources without gaining new information regarding the location of the globally optimal solution. From the computational viewpoint, it is important to avoid revisiting the same local optimally solution in order to maintain a high level of efficiency.